We present a simple model of diffusions coupled through their drifts in a
way that each component mean-reverts to the mean of the ensemble. In
particular, we are interested in the number of components reaching a
"default" level in a given time. This coupling creates stability of the
system in the sense that there is a large probability of "nearly no
default". However, we show that this "swarming" behavior also creates a
small probability that a large number of components default corresponding to
a "systemic risk event". The goal is to illustrate systemic risk with a toy
model of lending and borrowing banks, using mean-field limit and large
deviation estimates for a simple linear model. In the last part of the talk
we will show some recent work with Rene Carmona on a "Mean Field Game"
version of the previous model and the effects of the game on stability and
systemic risk.